There is no way to know for sure that Roosevelt's background (as opposed to his actual actions) helped build confidence among Americans. We can surmise, however, that Roosevelt's upper-class background helped to make him a credible person in the eyes of the general public.
FDR came from a rich and privileged background. He was able to do things like going on European trips as a child and he attended elite schools his entire life. In addition to being rich, he was from a prominent background, being related to President Theodore Roosevelt.
We can argue that people were reassured by this background. They might have felt that a person from such an elite background must have a good idea of what to do to solve America's problems. They might also have felt that there was no need to worry about him being too radical because someone from that background couldn't possibly go too far and do something like making the US a socialist country.